# MATHEMATICAL METHODS OF PHYSICS WITH EXERCISES

MODULE I: Basic concepts in the theory of Metric Spaces, and in the Calculus for functions of one real variable; elementary notions about partial derivatives. No preparatory constraint.

MODULE II: Basic elements of linear algebra, and of the theory of finite dimensional vector spaces

The final assessment consists in two examinations, at the end of each module, with the aim of verify the reached level of learning of the expected results.

MODULE I: The one assessment is the final written examination, which involves the solution of 4/5 questions/problems in 3/4 hours.

MODULE II: Written exam in which students must solve three exercises, each divided by two to four issues, of the same type as those performed in the classroom.

An oral test follows in which the student must demonstrate to have reached a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use, beyond the mere formal manipulation.

MODULE I: The course is meant to provide an introduction to Complex Analysis, or, more precisely, to the theory of functions of one complex variable. This is a fundamental subject, that naturally supplements the basic courses in Calculus, and is more or less relevant to all mathematicians, whatever their specialist inclination, and to physicists, whether applied or theoretical. Students will be led to realize that all basic functions of Calculus, originally introduced as functions of a real variable, are more naturally defined as functions of a complex variable; and that in this way a much deeper insight in their basic structure is gained. At the same time they will be introduced to some calculation tools based on complex analysis, which are of frequent use in applied mathematics and in physics. These include widely used techniques such as path integration, power series, and their use in the solution of ordinary differential equations.

MODULE II: It is expected that the students of this course will acquire a certain operational familiarity with those mathematical tools which, although they have long been commonly used in quantum mechanics, are nevertheless advanced enough to exceed the limits of basic courses in mathematical analysis; nevertheless maintaining a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use of it, beyond the pure and simple formal manipulation.

At the end of the course the students will be able to:

-recognize the analytic functions over the complex field, with their main properties;

-apply the residue theorem to compute integrals;

-recognize, and solve by power series methods, the main differential equations of mathematical physics;

-apply the complex analytic methods to the solution of problems often appearing in physics and in applied mathematics;

-know the main properties of the Hilbert and Banach spaces realized using the theory of measure;

-recognize the main properties of the bounded linear maps over a Hilbert space;

-apply the Fourier transform in L2, L1 spaces, and on the space of tempered distributions;

-use the expansion in Fourier series in order to compute the sum of notable series;

-recognize the basic properties of non linear operators on Hilbert spaces;

-recognize the tempered distributions, their principal operations, and their application to the determination of the fundamental solutions of linear partial differential equations.